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### Digital Logic Circuits

- Binary Logic and Gates
- Logic Simulation
- Boolean Algebra
- NAND/NOR and XOR gates
- Decoder fundamentals
- Half Adder, Full Adder, Ripple Carry Adder

Analog vs Digital

- Analog
- Continuous
- Time
- Every time has a value associated with it, not just some times
- Magnitude
- A variable can take on any value within a range
- e.g.
- temperature, voltage, current, weight, length, brightness, color

Digital Systems

Digital vs. Analog Waveforms

Digital:

only assumes discrete values

Analog:

values vary over a broad range

continuously

Analog vs Digital

- Digital
- Discontinuous
- Time (discretized)
- The variable is only defined at certain times
- Magnitude (quantized)
- The variable can only take on values from a finite set
- e.g.
- Switch position, digital logic, Dow-Jones Industrial, lottery, batting-average

Analog to Digital

- A Continuous Signal is Sampled at Some Time and Converted to a Quantized Representation of its Magnitude at that Time
- Samples are usually taken at regular intervals and controlled by a clock signal
- The magnitude of the signal is stored as a sequence of binary valued (0,1) bits according to some encoding scheme

Digital to Analog

- A Binary Valued, B = { 0, 1 }, Code Word can be Converted to its Analog Value
- Output of D/A Usually Passed Through Analog Low Pass Filter to Approximate a Continuous Signal
- Many Applications Construct a Signal Digitally and then D/A
- e.g., RF Transmitters, Signal Generators

Digital is Ubiquitous

- Electronic Circuits based on Digital Principles are Widely Used
- Automotive Engine/Speed Controllers
- Microwave Oven Controllers
- Heating Duct Controls
- Digital Watches
- Cellular Phones
- Video Games

Why Digital?

- Increased Noise Immunity
- Reliable
- Inexpensive
- Programmable
- Easy to Compute Nonlinear Functions
- Reproducible
- Small

Digital Design Process

- Computer Aided Design Tools
- Design entry
- Synthesis
- Verification and simulation
- Physical design
- Fabrication
- Testing

Representations for combinational logic

- Exclusive-or (XOR, EXOR, not-equivalence, ring-OR)
- Algebraic symbol:
- Gate symbol:

- Truth tables
- Graphical (logic gates)
- Algebraic equations (Boolean)

Representations of a Digital Design

Truth Tables

tabulate all possible input combinations and their associated

output values

Example: half adder

adds two binary digits

to form Sum and Carry

Example: full adder

adds two binary digits and

Carry in to form Sum and

Carry Out

NOTE: 1 plus 1 is 0 with a

carry of 1 in binary

Representations of Digital Design: Boolean Algebra

values: 0, 1

variables: A, B, C, . . ., X, Y, Z

operations: NOT, AND, OR, . . .

NOT X is written as X

X AND Y is written as X & Y, or sometimes X Y

X OR Y is written as X + Y

Deriving Boolean equations from truth tables:

Sum = A B + A B

Carry

0

0

0

1

A

0

0

1

1

B

0

1

0

1

Sum

0

1

1

0

OR\'d together product terms

for each truth table

row where the function is 1

if input variable is 0, it appears in

complemented form;

if 1, it appears uncomplemented

Carry = A B

Representations of a Digital Design: Boolean Algebra

Another example:

Sum = A B Cin + A B Cin + A B Cin + A B Cin

Sum

0

1

1

0

1

0

0

1

Cout

0

0

0

1

0

1

1

1

A

0

0

0

0

1

1

1

1

B

0

0

1

1

0

0

1

1

Cin

0

1

0

1

0

1

0

1

Cout = A B Cin + A B Cin + A B Cin + A B Cin

Gate Representations of a Digital Design

most widely used primitive building block in digital system design

Standard

Logic Gate

Representation

Half Adder Schematic

Net: electrically connected collection of wires

Netlist: tabulation of gate inputs & outputs

and the nets they are connected to

Elementary Binary Logic Functions

- Digital circuits represent information using two voltage levels.
- binary variables are used to denote these values
- by convention, the values are called “1” and “0” and we often think of them as meaning “True” and “False”
- Functions of binary variables are called logic functions.
- AND(A,B) = 1 if A=1 and B=1, else it is zero.
- AND is generally written in the shorthand A×B (or A&B or AÙB)
- OR(A,B) = 1 if A=1 or B=1, else it is zero.
- OR is generally written in the shorthand form A+B (or A|B or AÚB)
- NOT(A) = 1 if A=0 else it is zero.
- NOT is generally written in the shorthand form (or ØA or A)

- AND, OR and NOT can be used to express all other logic functions.

NAND

NOR

OR

BÞA

EXOR

A

ONE

ZERO

AND

(BÞA)

B

A

(AÞB)

AÞB

B

A

B

1111

0001

0000

1000

1100

0100

0010

0111

0101

1010

1110

1101

1011

0011

1001

0110

0011

0101

Two Variable Binary Logic Functions- Can make similar truth tables for 3 variable or 4 variable functions, but gets big (256 & 65,536 columns).

- Representing functions in terms of AND, OR, NOT.
- NAND(A,B) = (A×B)
- EXOR(A,B) = (A×B) + (A×B)

X

X

AND Gate

X×Y

Y

Y

X

X×Y

OR Gate

X+Y

Y

X+Y

Inverter

X

X’

X’

Basic Logic Gates- Logic gates “compute” elementary binary functions.
- output of an AND gate is “1” when both of its inputs are “1”, otherwise the output is zero
- similarly for OR gate and inverter
- Timing diagram shows how output values change over time as input values change

A

B

B

C

C

D

E

F

Multivariable Gates6 input OR Gate

3 input AND Gate

A+B+C+D+E+F

- AND function on n variables is “1” if and only if ALL its arguments are “1”.
- n input AND gate output is “1” if all inputs are “1”
- OR function on n variables is “1” if and only if at least one of its arguments is “1”.
- n input OR gate output is “1” if any inputs are “1”
- Can construct “large” gates from 2 input gates.
- however, large gates can be less expensive than required number of 2 input gates

A×B×C

B

C

B×C

A+B×C

00001111

00110011

01010101

00100010

00101111

A

B

A+B×C

C

Elements of Boolean Algebra- Boolean algebra defines rules for manipulating symbolic binary logic expressions.
- a symbolic binary logic expression consists of binary variables and the operators AND, OR and NOT (e.g. A+B×C)
- The possible values for any Boolean expression can be tabulated in a truth table.

- Can define circuit forexpression by combininggates.

Schematic Capture & Logic Simulation

wires

advancesimulation

gates

signalwaveforms

terminals

schematicentry tools

signalnames

A+B×C

(B×(C))

Boolean Functions to Logic Circuits- Any Boolean expression can be converted to a logic circuit made up of AND, OR and NOT gates.

step 1: add parentheses to expression to fully define order of operations - A+(B×(C))

step 2: create gate for “last” operation in expression

gate’s output is value of expression

gate’s inputs are expressions combined by operation

step 3: repeat for sub-expressions and continue until done

- Number of simple gates needed to implement expression equals number of operations in expression.
- so, simpler equivalent expression yields less expensive circuit
- Boolean algebra provides rules for simplifying expressions

Basic Identities of Boolean Algebra

2. X×1 =X

4. X×0 = 0

6. X×X=X

8. X×X’ = 0

11. X×Y=Y×X

13. X×(Y×Z) = (X×Y)×Z

15. X+(Y×Z) =(X+Y)×(X+Z)

17. (X×Y)’ = X+Y

1. X + 0 =X

3. X + 1 = 1

5. X + X=X

7. X + X’ = 1

9. (X’)’ =X

10. X + Y=Y + X

12. X+(Y+Z) = (X+Y)+Z

14. X(Y+Z) =X×Y + X×Z

16. (X+ Y)=X×Y

commutative

associative

distributive

DeMorgan’s

- Identities define intrinsic properties of Boolean algebra.
- Useful in simplifying Boolean expressions
- Note: 15-17 have no counterpart in ordinary algebra.
- Parallel columns illustrate duality principle.

(X+ Y)=X×Y

XYZ

Y×Z

X+(Y×Z)

X+Y

X+Z

(X+Y)×(X+Z)

XY

(X+ Y)

X×Y

00

1

1

000

0

0

0

0

0

0

01

0

0

1

0

0

0

001

0

10

0

0

0

010

0

1

0

0

11

0

1

011

1

1

1

1

1

1

1

100

0

1

1

101

0

1

1

1

0

110

1

1

1

1

1

111

1

1

1

1

Verifying Identities Using Truth Tables- Can verify any logical equation with small number of variables using truth tables.
- Break large expressions into parts, as needed.

DeMorgan’s Laws for n Variables

- We can extend DeMorgan’s laws to 3 variables by applying the laws for two variables.

(X + Y + Z)=(X+ (Y + Z))- by associative law

=X×(Y + Z)- by DeMorgan’s law

=X×(Y×Z) - by DeMorgan’s law

= X×Y×Z- by associative law

(X×Y×Z)=(X×(Y×Z))- by associative law

=X+ (Y×Z) - by DeMorgan’s law

=X+ (Y+ Z)- by DeMorgan’s law

= X+ Y+ Z- by associative law

- Generalization to n variables.
- (X1 + X2 + × × × + Xn)=X1×X2× × × Xn
- (X1×X2× × × Xn)=X1 + X2 + × × × + Xn

Y

Z

by identity 14

F=XY(Z +Z)+XZ

by identity 7

X

Y

F=XY×1+XZ =XY +XZ by identity 2

X

Z

Y

Z

Simplification of Boolean ExpressionsF=XYZ+XYZ+XZ

The Duality Principle

- The dual of a Boolean expression is obtained by interchanging all ANDs and ORs, and all 0s and 1s.
- example: the dual of A+(B×C)+0 is A×(B+C)×1
- The duality principle states that if E1 and E2 are Boolean expressions then

E1= E2 dual(E1)=dual(E2)

where dual(E) is the dual of E. For example,

A+(B×C)+0 = (B×C)+D A×(B+C)×1 = (B+C)×D

Consequently, the pairs of identities (1,2), (3,4), (5,6), (7,8), (10,11), (12,13), (14,15) and (16,17) all follow from each other through the duality principle.

The Consensus Theorem

Theorem. XY + XZ +YZ = XY + XZ

Proof. XY + XZ +YZ = XY + XZ + YZ(X + X) 2,7

= XY + XZ + XYZ + XYZ 14

= XY + XYZ + XZ + XYZ 10

= XY(1 + Z)+ XZ(1 + Y) 2,14

= XY + XZ 3,2

Example. (A + B)(A+ C) = AA+ AC + AB + BC

= AC + AB + BC

= AC + AB

Dual. (X + Y)(X + Z)(Y + Z) = (X + Y)(X + Z)

Taking the Complement of a Function

Method 1. Apply DeMorgan’s Theorem repeatedly.

(X(YZ+ YZ)) = X+ (YZ+ YZ)

= X + (YZ)(YZ)

= X+ (Y + Z)(Y+ Z)

Method 2. Complement literals and take dual

(X(YZ+ YZ))= dual(X(YZ + YZ))

= X+ (Y + Z)(Y+ Z)

Sum of Products Form

- The sum of products is one of two standard forms for Boolean expressions.

sum-of-products-expression = term+term ... +term

term = literal×literal×××× ×literal

Example. XYZ + XZ + XY + XYZ

- A minterm is a term that contains every variable, in either complemented or uncomplemented form.

Example. in expression above, XYZ is minterm, but XZ is not

- A sum of minterms expression is a sum of products expression in which every term is a minterm

Example.XYZ + XYZ + XYZ + XYZ is sum of minterms expression that is equivalent to expression above

Product of Sums Form

- The product of sums is the second standard form for Boolean expressions.

product-of-sums-expression = s-term×s-term ... ×s-term

s-term = literal+literal+××× +literal

Example. (X+Y+Z )(X+Z)(X+Y)(X+Y+Z)

- A maxterm is a sum term that contains every variable, in complemented or uncomplemented form.

Example. in exp. above, X+Y+Z is a maxterm, but X+Z is not

- A product of maxterms expression is a product of sums expression in which every term is a maxterm

Example.(X+Y+Z )(X+Y+Z)(X+Y+Z)(X+Y+Z)is product of maxterms expression that is equivalent to expression above

X

NAND Gate

NOR Gate

(X+Y)

(X×Y)

Y

Y

=

=

=

=

NAND and NOR Gates- In certain technologies (including CMOS), a NAND (NOR) gate is simpler & faster than an AND (OR) gate.
- Consequently circuits are often constructed using NANDs and NORs directly, instead of ANDs and ORs.
- Alternative gate representations makes this easier.

A

EXOR gate

A

AB+AB

B

B

Exclusive Or and Odd Function- The odd function on n variables is 1 when an odd number of its variables are 1.
- odd(X,Y,Z) = XYZ+ XYZ + XYZ + XYZ = X Y Z
- similarly for 4 or more variables
- Parity checking circuits use the odd function to provide a simple integrity check to verify correctness of data.
- any erroneous single bit change will alter value of odd function, allowing detection of the change

- The EXOR function is defined by AB = AB + AB.

X

X+Y

X×Y

Y

Y

Positive and Negative Logic- In positive logic systems, a high voltage is associated with a logic 1, and a low voltage with a logic 0.
- positive logic is just one of two conventions that can be used to associate a logic value with a voltage
- sometimes it is more convenient to use the opposite convention
- In logic diagrams that use negative logic, a polarity indicator is used to indicate the correct logical interpretation for a signal.

- Circuits commonly use a combination of positive and negative logic.

Decoder Fundamentals

- Route data to one specific output line.
- Selection of devices, resources
- Code conversions.
- Arbitrary switching functions
- implements the AND plane
- Asserts one-of-many signal; at most one output will be asserted for any input combination

Encoding

Binary

Decimal Unencoded Encoded

0 0001 00

1 0010 01

2 0100 10

3 1000 11

Note: Finite state machines may be unencoded ("one-hot")

or binary encoded. If the all 0\'s state is used, then

one less bit is needed and it is called modified

one-hot coding.

B

1 1

1 0

0 1

00

Y

Y

Y

Y

E Q 3

E Q 2

E Q 1

E Q 0

AND 2

A

B

AND 2 A

A

B

AND 2 A

D 0

A

B

AND 2 B

D 1

2:4 DecoderWhat happens when the inputs goes from 01 to 10?

B

C

A

B

C

A

B

C

A

B

C

1 1

1 0

0 1

00

Y

Y

Y

Y

E Q 3

E Q 2

E Q 1

E Q 0

AND 3

AND 3 A

AND 3 A

D 0

D 1

ENABLE

AND 3 B

2:4 Decoder with Enable1 1

1 0

0 1

00

One-bit Full Adder (FA)

3 inputs: A, B, C

2 outputs: S, Co

Truth table:

Schematic View:

cell-based approach

Full Adder – with EXOR and NANDA, B, C S, Co

0, 0, 0 0, 0

0, 0, 1 1, 0

0, 1, 0 1, 0

0, 1, 1 0, 1

1, 0, 0 1, 0

1, 0, 1 0, 1

1, 1, 0 0, 1

1, 1, 1 1, 1

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